{smcl}
{* 12dec2007}{...}
{cmd:help mata mm_benford()}
{hline}

{title:Title}

{p 4 14 2}
{bf:mm_benford() -- Benford's Law (distribution of digits)}

{title:Syntax}

{p 8 23 2}
{it:real vector}{space 1}
{cmd:mm_benford(}{it:digits} [{cmd:,} {it:position}{cmd:,} {it:base}]{cmd:)}


{pstd}
where

{p 7 16 2}
{it:digits}:  {it:real vector} providing the digits for which the probabilities 
be computed; the digits must be in {c -(}0,...,{it:base}-1{c )-}

{p 5 16 2}
{it:position}:  {it:real scalar} specifying the position of the digits

{p 9 16 2}
{it:base}:  {it:real scalar} specifying the base of the numeric system


{title:Description}

{pstd}
{cmd:mm_benford()} returns the probabilities of the specified {it:digits} according 
to Benford's Law (Newcomb 1881, Benford 1938, Hill 1998). The formula for the 
1st (leading) digit in the base 10 number system is:

        P(d) = log10(1 + 1/d)

{pstd}
with d in {c -(}1,...,9{c )-}.

{pstd}The general formula for the p-th digit in the base b number system is
                                         
                           b^(p-1) - 1
                              {hline 3}
                       1      \      {c TLC}      1   {c TRC} 
        P(d, p, b) = {hline 5}         ln{c |}1 + {hline 5} {c |}
                     ln(b)    /      {c BLC}    k*b+d {c BRC} 
                              {hline 3}
                           k = b^(p-2)

{pstd}
with d in {c -(}0,...,b-1{c )-} (see, e.g., 
{browse "http://www.mathpages.com/home/kmath302/kmath302.htm"}). 

{pstd}The limit of 
P(d) with rising p is 1/b. Computation becomes exceedingly 
intensive with rising p since the number of operations is in the order
of b^p - b^(p-1).

{title:Remarks}

{pstd} Examples

{pstd} Fist digit Benford distribution:

        {com}: (1::9), mm_benford(1::9)
        {res}       {txt}          1             2
            {c TLC}{hline 29}{c TRC}
          1 {c |}  {res}          1   .3010299957{txt}  {c |}
          2 {c |}  {res}          2   .1760912591{txt}  {c |}
          3 {c |}  {res}          3   .1249387366{txt}  {c |}
          4 {c |}  {res}          4    .096910013{txt}  {c |}
          5 {c |}  {res}          5    .079181246{txt}  {c |}
          6 {c |}  {res}          6   .0669467896{txt}  {c |}
          7 {c |}  {res}          7    .057991947{txt}  {c |}
          8 {c |}  {res}          8   .0511525224{txt}  {c |}
          9 {c |}  {res}          9   .0457574906{txt}  {c |}
            {c BLC}{hline 29}{c BRC}{txt}
        
{pstd} Second digit Benford distribution:

        {com}: (0::9), mm_benford(0::9,2)
        {res}        {txt}          1             2
             {c TLC}{hline 29}{c TRC}
           1 {c |}  {res}          0   .1196792686{txt}  {c |}
           2 {c |}  {res}          1   .1138901034{txt}  {c |}
           3 {c |}  {res}          2    .108821499{txt}  {c |}
           4 {c |}  {res}          3   .1043295602{txt}  {c |}
           5 {c |}  {res}          4   .1003082023{txt}  {c |}
           6 {c |}  {res}          5   .0966772358{txt}  {c |}
           7 {c |}  {res}          6   .0933747358{txt}  {c |}
           8 {c |}  {res}          7   .0903519893{txt}  {c |}
           9 {c |}  {res}          8   .0875700536{txt}  {c |}
          10 {c |}  {res}          9   .0849973521{txt}  {c |}
             {c BLC}{hline 29}{c BRC}{txt}

{pstd} Second digit Benford distribution in base 6:

        {com}: (0::5), mm_benford(0::5,2,6)
        {res}       {txt}          1             2
            {c TLC}{hline 29}{c TRC}
          1 {c |}  {res}          0   .2019648465{txt}  {c |}
          2 {c |}  {res}          1   .1841218513{txt}  {c |}
          3 {c |}  {res}          2   .1697091954{txt}  {c |}
          4 {c |}  {res}          3   .1577441976{txt}  {c |}
          5 {c |}  {res}          4   .1476009617{txt}  {c |}
          6 {c |}  {res}          5   .1388589474{txt}  {c |}
            {c BLC}{hline 29}{c BRC}{txt}
    

{title:Conformability}

    {cmd:mm_benford(}{it:digits}{cmd:,} {it:position}{cmd:,} {it:base}{cmd:)}:
    {it:digits}:  {it:r x} 1 or 1 {it:x c}
  {it:position}:  1 {it:x} 1
      {it:base}:  1 {it:x} 1
    {it:result}:  {it:r x} 1 or 1 {it:x c}.
   

{title:Diagnostics}

{p 4 4 2}{cmd:mm_benford(0,1,}{it:base}{cmd:)} evaluates to missing.


{title:Source code}

{p 4 4 2}
{help moremata_source##mm_benford:mm_benford.mata}


{title:References}

{phang}
Benford, Frank (1938). The law of anomalous numbers. Proceedings of the 
American Philosophical Society 78: 551-572.

{phang}
Hill, Theodore P. (1998). The first digit phenomenon. American Scientist 86: 358. 

{phang}
Newcomb, Simon (1881). Note on the frequency of use of the different 
digits in natural numbers. American Journal of Mathematics 4: 39-40.
{p_end}


{title:Author}

{p 4 4 2} Ben Jann, University of Bern, jann@soz.unibe.ch


{title:Also see}

{p 4 13 2}
Online:  help for
{bf:{help m4_statistical:[M-4] statistical}},
{bf:{help moremata}}
{p_end}






